The sets $A = \{z : z^{18} = 1\}$ and $B = \{w : w^{48} = 1\}$ are both sets of complex roots of unity. The set $C = \{zw : z \in A \ \text{and} \ w \in B\}$ is also a set of complex roots of unity. How many distinct elements are in $C$?

Prove that $$ (n+2)^n=\prod_{k=1}^{n+1} \sum_{l=1}^{n+1} le^{\frac{2i\pi k (n-l+1)}{n+2}} , $$ for any natural number $ n. $ [i]Mihai Piticari[/i]

Let incircle $(I)$ of triangle $ABC$ touch the sides $BC,CA,AB$ at $D,E,F$ respectively. Let $(O)$ be the circumcircle of $ABC$. Ray $EF$ meets $(O)$ at $M$. Tangents at $M$ and $A$ of $(O)$ meet at $S$. Tangents at $B$ and $C$ of $(O)$ meet at $T$. Line $TI$ meets $OA$ at $J$. Prove that $\angle ASJ=\angle IST$.

Let $z_1$ and $z_2$ be complex numbers. Prove that \[|z_1+z_2|+|z_1-z_2|\leqslant |z_1|+|z_2|+\max\{|z_1|,|z_2|\}.\][i]Vlad Cerbu and Sorin Rădulescu[/i]

To be considered the following complex and distinct $a,b,c,d$. Prove that the following affirmations are equivalent: i)For every $z\in \mathbb{C}$ the inequality takes place :$\left| z-a \right|+\left| z-b \right|\ge \left| z-c \right|+\left| z-d \right|$; ii)There is $t\in \left( 0,1 \right)$ so that $c=ta+\left( 1-t \right)b$ si $d=\left( 1-t \right)a+tb$

Let $ABC$ be an acute triangle with $\angle BAC=30^{\circ}$. The internal and external angle bisectors of $\angle ABC$ meet the line $AC$ at $B_1$ and $B_2$, respectively, and the internal and external angle bisectors of $\angle ACB$ meet the line $AB$ at $C_1$ and $C_2$, respectively. Suppose that the circles with diameters $B_1B_2$ and $C_1C_2$ meet inside the triangle $ABC$ at point $P$. Prove that $\angle BPC=90^{\circ}$ .

The following sets of points are considered in the plane: $A=\{$ affixes of complexes $z$ such that arg $(z - (2 + 3i))=\pi /4\}$, $B =\{$ affixes of complexes $z$ such that mod $( z- (2 + i)<2\}$. Determine the orthogonal projection on the $X$ axis of $A \cap B$.

Let $ABCDEF$ be a convex hexagon with $AB=DE$, $BC=EF$, $CD=FA$, and $\angle A-\angle D = \angle C -\angle F = \angle E -\angle B$. Prove that the diagonals $AD$, $BE$, and $CF$ are concurrent.

Let $ABC$ be a triangle and $A_1$, $B_1$, $C_1$ the midpoints of the sides $BC$, $CA$ and $AB$ respectively. Prove that if $M$ is a point in the plane of the triangle such that \[ \frac{MA}{MA_1} = \frac{MB}{MB_1} = \frac{MC}{MC_1} = 2 , \] then $M$ is the centroid of the triangle.

Let $a \ge1$ be a real number and $z$ be a complex number such that $| z + a | \le a$ and $|z^2+ a | \le a$. Show that $| z | \le a$.

Let $\zeta= e^{2\pi i/99}$ and $\omega e^{2\pi i/101}$. The polynomial $$x^{9999} + a_{9998}x^{9998} + ...+ a_1x + a_0$$ has roots $\zeta^m + \omega^n$ for all pairs of integers $(m, n)$ with $0 \le m < 99$ and $0 \le n < 101$. Compute $a_{9799} + a_{9800} + ...+ a_{9998}$.

Give two statements: (1) $a,b,c$ are complex numbers, if $a^2+b^2>c^2$, then $a^2+b^2-c^2>0$. (2) $a,b,c$ are complex numbers, if $a^2+b^2-c^2>0$, then $a^2+b^2>c^2$. Then, which is true? $\text{(A)}$ (1) is correct, (2) is correct as well $\text{(B)}$ (1) is correct, (2) is incorrect $\text{(C)}$ (1) is incorrect, (2) is incorrect as well $\text{(D)}$ (1) is incorrect, (2) is correct

$z$ is a complex number and $|z|=1$ and $z^2\ne1$. Then $\frac z{1-z^2}$ lies on $\textbf{(A)}~\text{a line not through origin}$ $\textbf{(B)}~\text{|z|=2}$ $\textbf{(C)}~x-\text{axis}$ $\textbf{(D)}~y-\text{axis}$

Consider the equation $10z^2-3iz-k=0$, where $z$ is a complex variable and $i^2=-1$. Which of the following statements is true? $ \textbf{(A)}\ \text{For all positive real numbers}\ k,\ \text{both roots are pure imaginary.} \\ \qquad\textbf{(B)}\ \text{For all negative real numbers}\ k,\ \text{both roots are pure imaginary.} \\ \qquad\textbf{(C)}\ \text{For all pure imaginary numbers}\ k,\ \text{both roots are real and rational.} \\ \qquad\textbf{(D)}\ \text{For all pure imaginary numbers}\ k,\ \text{both roots are real and irrational.} \\ \qquad\textbf{(E)}\ \text{For all complex numbers}\ k,\ \text{neither root is real.} $

We consider a natural number $n$, $n \geq 2$ and the matrices \begin{tabular}{cc} $A= \begin{pmatrix} 1 & 2 & 3 & \cdots & n\\ n & 1 & 2 & \cdots & n - 1\\ n - 1 & n & 1 & \cdots & n - 2\\ \cdots & \cdots & \cdots & \cdots & \cdots\\2 & 3 & 4 & \cdots & 1 \end{pmatrix}$ \end{tabular} Show that$$\epsilon^ndet\left(I_n-A^{2n}\right)+\epsilon^{n-1}det\left(\epsilon I_n-A^{2n}\right)+\epsilon^{n-2}det\left(\epsilon^2 I_n-A^{2n}\right)+\cdots +det\left(\epsilon^n I_n-A^{2n}\right)$$ $$=n(-1)^{n-1}\left[\frac{n^n(n+1)}{2}\right]^{2n^2-4n}\left(1+(n+1)^{2n}\left(2n+(-1)^n{{2n}\choose{n}}\right)\right)$$where $\epsilon \in \mathbb{C}\backslash \mathbb{R}$, $\epsilon^{n+1}=1$

The complex numbers $z_1$ and $z_2$ are given such that $z_1=-1+i$ and $z_2=2+4i$. Find the complex number $z_3$ such that $z_1,z_2,z_3$ are the points of an equilateral triangle. How many solutions do we have ?

Let be two complex numbers $ a,b. $ Show that the following affirmations are equivalent: $ \text{(i)} $ there are four numbers $ x_1,x_2,x_3,x_4\in\mathbb{C} $ such that $ \big| x_1 \big| =\big| x_3 \big|, \big| x_2 \big| =\big| x_4 \big|, $ and $$ x_{j_1}^2-ax_{j_1}+b=0=x_{j_2}^2-bx_{j_2}+a,\quad\forall j_1\in\{ 1,2\} ,\quad\forall j_2\in\{ 3,4\} . $$ $ \text{(ii)} a^3=b^3 $ or $ b=\overline{a} $ (the conjugate of a).

Let $z_1,z_2$ and $z_3$ be complex numbers of modulus $1,$ such that $|z_i-z_j|\geq\sqrt{2}$ for all $i\neq j\in\{1,2,3\}.$ Prove that \[|z_1+z_2|+|z_2+z_3|+|z_3+z_2|\leq 3.\][i]Mathematical Gazette[/i]

Let $z=x+yi$ be a complex number with $x$ and $y$ rational and with $|z|=1.$ Prove that the number $|z^{2n} -1|$ is rational for every integer $n$.

(a) Let $z_1,z_2$ be complex numbers such that $z_1z_2=1$ and $|z_1-z_2|=2$. Let $A,B,M_1,M_2$ denote the points in complex plane corresponding to $-1,1,z_1,z_2$, respectively. Show that $AM_1BM_2$ is a trapezoid and compute the lengths of its non-parallel sides. Specify the particular cases. (b) Let $\mathcal C_1$ and $\mathcal C_2$ be circles in the plane with centers $O_1$ and $O_2$, respectively, and with radius $d\sqrt2$, where $2d=O_1O_2$. Let $P$ and $Q$ be two variable points on $\mathcal C_1$ and $\mathcal C_2$ respectively, both on $O_1O_2$ on on different sides of $O_1O_2$, such that $PQ=2d$. Prove that the locus of midpoints $I$ of segments $PQ$ is the same as the locus of points $M$ with $MO_1\cdot MO_2=m$ for some $m$.

Given $P(z)= z^2 +az +b,$ where $a,b \in \mathbb{C}.$ Suppose that $|P(z)|=1$ for every complex number $z$ with $|z|=1.$ Prove that $a=b=0.$

The complex numbers $z$ and $w$ satisfy the system \begin{align*}z+\frac{20i}{w}&=5+i,\\w+\frac{12i}{z}&=-4+10i.\end{align*} Find the smallest possible value of $|zw|^2$.

Let $\omega$ be a complex number satisfying $\omega^{2048} = 1$ and $\omega^{1024} \neq 1$. Find the unique ordered pair of nonnegative integers $(p, q)$ satisfying \[ 2^p - 2^q = \sum_{0 \leq m < n \leq 2047} (\omega^m + \omega^n)^{2048}. \]

Let $n\geqslant 2$ be an integer. Determine all complex numbers $z{}$ which satisfy \[|z^{n+1}-z^n|\geqslant|z^{n+1}-1|+|z^{n+1}-z|.\]

Let $ p(x) \equal{} x^3 \plus{} ax^2 \plus{} bx \plus{} c$, where $ a$, $ b$, and $ c$ are complex numbers. Suppose that \[ p(2009 \plus{} 9002\pi i) \equal{} p(2009) \equal{} p(9002) \equal{} 0 \]What is the number of nonreal zeros of $ x^{12} \plus{} ax^8 \plus{} bx^4 \plus{} c$? $ \textbf{(A)}\ 4\qquad \textbf{(B)}\ 6\qquad \textbf{(C)}\ 8\qquad \textbf{(D)}\ 10\qquad \textbf{(E)}\ 12$